Definition 1.1.1.
In the context of geometry, ordinary numbers (integers, rational numbers or real numbers) are called scalars.
Section 1.1 Vectors in \(\RR^2\) and \(\RR^3\)
Subsection 1.1.1 Describing Space
The purpose of vectors is to describe the geometry of space. For the purposes of this course, I means two and three dimensional space, but many of the tools are more general and can describe higher dimensional spaces. That’s very interesting for a more abstract treatment, but to do multivariable calculus, I need two and three dimensional vectors.
I imagine this is not your first exposure to vectors. However, this week is nevertheless important to review the main ideas as well as set our conventions for notation and terminology. Vectors are used in many places (linear algebra, physics, computer graphcis, engineering, among others) and most of those places have minor or even major differences in terms and notations. This review will get us all on the same page for the discussions through the course. So, let me start with the basic definitions.
Definition 1.1.2.
A vector is a finite ordered list of scalars. Vectors can be written either as columns or rows. If a vector is a list of the numbers \(4\text{,}\) \(15\text{,}\) \(\pi\) and \(-e\) (in that order), I write the vector in one of two ways:
\begin{align*}
\amp \begin{pmatrix} 4 \\ 15 \\ \pi \\ -e \end{pmatrix}
\amp \text{or} \amp \amp (4, 15, \pi, -e) \amp
\end{align*}
If you have taken my linear algebra course, you will know that, in that course, I almost exclusively use column vectors. Here the convention is reversed: I will almost exclusively use row vectors.
Definition 1.1.3.
Let \(n\) be a positive integer. Real Euclidean Space or Cartesian Space, written \(\RR^n\text{,}\) is the set of all vectors of length \(n\) with real number entries.
Since there are \(n\) coefficieint in its vectors, the space \(\RR^n\) has dimension \(n\text{.}\)
In this course, I am exclusively interested in \(\RR^2\) (the familiar Cartesian Plane) and in \(\RR^3\) (which I will call Cartesian Three-Space or, more briefly, just Cartesian Soace).
Definition 1.1.4.
If I have a vector \((a,b)\) in \(\RR^2\) or \((a,b,c)\) in \(\RR^3\text{,}\) the scalars \(a\text{,}\) \(b\) and \(c\) are called the entries, coordinates or components of that vector. Specifically, \(a\) is the first component, \(b\) is the second component and \(c\) is the third component.
Definition 1.1.5.
The vector \((0,0)\) in \(\RR^2\) and the vector \((0,0,0)\) in \(\RR^3\) are called the origin of each space. Each is also called the zero vector.
Euclidean space is visualized by drawing axes, one in each independent perpendicular direction. In this visualization, the vector \((a,b)\) corresponds to the unique point I get moving \(a\) units in the direction of the \(x\) axis and \(b\) units in the direction of the \(y\) axis. Figure 1.1.6 shows the location of several points in \(\RR^2\text{.}\)
As with \(\RR^2\text{,}\) the point \((a,b,c)
\in \RR^3\) is the unique point I find by moving \(a\) units in the \(x\) direction, \(b\) units in the \(y\) direction and \(c\) units in the \(z\) direction. When visualizing \(\RR^2\text{,}\) the convention is to draw the \(x\) axis horizontally, with a positive direction to the right, and the \(y\) axis vertically, with a positive direction upwards. For \(\RR^3\text{,}\) the \(x\) and \(y\) axes form a flat plane, and the \(z\) axis extends vertically from that plane as shown in Figure 1.1.7.
Subsection 1.1.2 Points or Directions?
I can think of an element of \(\RR^2\text{,}\) say \((1,4)\text{,}\) as both the point located at \((1,4\) and the vector drawn from the origin to the point \((1,4)\text{,}\) as shown in Figure 1.1.8. Though these two ideas are distinct, I will frequently switch between them, often without explicitly saying so. Part of becoming proficient in vector geometry is becoming accustomed to the switch between the perspectives of points and directions.
Subsection 1.1.3 Local Direction Vectors
I’ve already spoken about the distinction between elements of \(\RR^n\) as points and vectors. There is another important subtlety that shows up all throughout vector geometry. In addition to thinking of vectors as directions starting at the origin, I can think of them as directions starting anywhere in \(\RR^n\text{.}\) I call these local direction vectors.
Figure 1.1.9 shows local direction vectors starting at the point \((2,2) \in \RR^2\text{.}\) The two local vectors \((1,0)\) and \((0,1)\) are relative to the point \((2,2) \in \RR^2\) as if that were their origin.
Using vectors to define local directions is a useful tool. A standard example is a camera in a three-dimensional virtual environment. First, I need to know the location of the camera, which is an ordinary vector starting from the origin. Second, I need to know what direction the camera is pointing, which is a local direction vector which treats the camera location as the current origin.
One of the most difficult things about learning vector geometry is becoming accustomed to local direction vectors. I won’t always carefully distinguish between vectors at the origin and local direction vectors; often, the difference is implied, and it is up to the reader/student to figure out how the vectors are being used.
